Pares básicos de Transformadas de Laplace ƒ(t) F(s)

Pares básicos de Transformadas de Laplace
ƒ(t)
F(s)
! (t)
1
d n! (t)
dt
sn
n
1
s
u(t)
1
t u(t)
s2
n!
t n u(t)
e!at u(t),
s n+1
1
s+a
a>0
t n e!at u(t),
n!
a>0
(s + a)n+1
!
s +!2
sen(! t) u(t)
2
s
cos(! t) u(t)
2
s +!2
s sen ! + " cos!
sen(! t + " ) u(t)
s2 + " 2
s cos! " # sen !
cos(! t + " ) u(t)
s2 + # 2
c s+ d!
s2 + ! 2
d&
#
2
2
c + d cos % ! t " arctan ( u(t)
$
c'
2
e!at sen(" t) u(t),
a>0
e!at cos(" t) u(t),
a>0
e!at sen(" t + # ) u(t),
a>0
e!at cos(" t + # ) u(t),
a>0
2 ! at
c +d e
d&
#
cos % " t ! arctan ( u(t),
$
c'
!
(s + a)2 + ! 2
s+a
(s + a)2 + ! 2
(s + a) sen ! + " cos!
(s + a)2 + " 2
(s + a) cos! " # sen !
(s + a)2 + # 2
a>0
c (s + a) + d !
(s + a)2 + ! 2
Propiedades de la Transformada de Laplace
ƒ(t)
F(s)
a1 ƒ1 (t) ± a2 ƒ 2 (t)
a1F1 (s) ± a2 F2 (s)
d
ƒ(t)
dt
sF(s) ! ƒ(0 ! )
d2
dt
2
dn
dt n
g(t) =
ƒ(t)
ƒ(t)
#
$0
ƒ(! )d! +g(0 " )
"
s 2 F(s) ! sƒ(0 ! ) ! ƒ '(0 ! )
s n F(s) ! s n!1 ƒ(0 ! ) ! s n!2
dƒ !
d n!1 ƒ
(0 ) ! L ! n!1 (0 ! )
dt
dt
F(s) g(0 ! )
+
s
s
u(t ! a)
e!as
s
ƒ(t ! a)u(t ! a)
e!as F(s)
e!at ƒ(t)
F(s + a)
ƒ(at)
1 ! s$
F# &
a " a%
t ƒ(t)
!
n
t ƒ(t)
(!1)
ƒ(t )
t
#s
lim ƒ(t)
t!0
lim ƒ(t)
t!"
t
#0 ƒ(! ) g(t " ! ) d!
d
F(s)
ds
dn
n
"
ds n
F(s)
F(! )d !
lim sF(s)
s!"
lim sF(s)
s!0
F(s)G(s )